Suppose that we have totally ordered sets (u)u(u,<) and (v)v(v,\prec) and words f:u→Anormal-:fnormal-→uAf\colon u\to A
and g:v→Anormal-:gnormal-→vAg\colon v\to A. Let u⁢∐vucoproductvu\coprod v denote the disjoint union of uuu and vvv and let
p:u→u⁢∐vnormal-:pnormal-→uucoproductvp\colon u\to u\coprod v and q:u→u⁢∐vnormal-:qnormal-→uucoproductvq\colon u\to u\coprod v be the canonical maps. Then
we may define an order ≪much-less-than\ll on u⁢∐vucoproductvu\coprod v as follows:

If x∈uxux\in u and y∈uyuy\in u, then p⁢(x)≪p⁢(y)much-less-thanpxpyp(x)\ll p(y) if and only if x<yxyx<y.

If x∈uxux\in u and y∈vyvy\in v, then p⁢(x)≪q⁢(y)much-less-thanpxqyp(x)\ll q(y).

If x∈vxvx\in v and y∈vyvy\in v, then q⁢(x)≪q⁢(y)much-less-thanqxqyq(x)\ll q(y) if and only if x≺yprecedesxyx\prec y.

We define the concatenation of fff and ggg, which will be denoted f∘gfgf\circ g, to be
map from u⁢∐vucoproductvu\coprod v to AAA defined by the following conditions: